The Cosine Rule

The formula sheet gives both forms of the Cosine Rule.

Which form is best suited for finding the length of a missing side?

Pay attention to where angle \(A\) fits into the formula when calculating side \(a\).

\(32.45\) cm

The formula sheet gives both forms of the Cosine Rule.

Which form is best suited for finding the size of a missing angle?

Pay attention to where side \(a\) fits into the formula when calculating angle \(A\).

\(68.0^{\circ}\)


Algebraic Fractions

Simplfying algebraic fractions should begin by factorising.

In an exam question, expect both the numerator and denominator to share the same factor

\(\dfrac{x-5}{x-3}\)

To add or subtract fractions, the denominators should be the same.

\(\dfrac{13x+7}{15}\)

The lowest common multiple of \(x+3\) and \(x-2\) is \((x+3)(x-2)\).

Start by rewriting each fraction with \((x+3)(x-2)\) as the denominator, adjusting the numerators appropriately.

\(\dfrac{7x+11}{(x+3)(x-2)}\)

The lowest common multiple of \(3x\) and \(x^2\) is \(3x^2\).

\(\dfrac{5x-6}{3x^2}\)

Approach algebraic fractions in a similar manner to numerical fractions.

What is the approach for divisions?

Then use cross-cancelling to simplify.

\(\dfrac{k}{6p}\)


Trig Equations

First rearrange the equation to the form \(\sin{x^{\circ}}=\dots\)

Remember to use the \(ASTC\) diagram to find two solutions.

\(x=53.1,126.9\)

First rearrange the equation to the form \(\tan{x^{\circ}}=\dots\)

Remember to use the \(ASTC\) diagram to find two solutions.

\(x=77.5,257.5\)

First rearrange the equation to the form \(\cos{x^{\circ}}=\dots\)

Remember to use the \(ASTC\) diagram to find two solutions.

Which two quadrants apply when \(\cos\) is negative?

\(x=104.5,255.5\)

First rearrange the equation to the form \(\sin{x^{\circ}}=\dots\)

Remember to use the \(ASTC\) diagram to find two solutions.

Which two quadrants apply when \(\sin\) is negative?

\(x=221.8,318.2\)

First rearrange the equation to the form \(\tan{x^{\circ}}=\dots\)

Remember to use the \(ASTC\) diagram to find two solutions.

\(x=53.1,233.1\)


Quadratics II

A number of approaches to completing the square question this are possible.

Check your notes and examples to see which method your teacher has taught.

\((x+6)^2-2\)

A number of approaches to completing the square question this are possible.

Check your notes and examples to see which method your teacher has taught.

\((x-4)^2+5\)

The turning point for \(y=(x-a)^2+b\) is \((a,b)\).

\((-5,3)\)

Start by writing the equation of the parabola using \((4,-3)\).

Compare this question to the previous one.

For part (c), any parabola with a turning point at \((a,b)\) has a vertical line (axis) of symmetry with the equation \(x=a\).

\(a=-4\), \(b=-3\) and axis of symmetry has equation \(x=4\).

To determine the nature of the roots, use the discriminant, \(b^2-4ac\).

Begin by stating the values of \(a\), \(b\) and \(c\).

Compare the result to zero to make a conclusion about the roots.

Pay close attention to the words which must be used in the answer.

\(44>0\) so real, distinct roots.

Note that each of the three possible conclusions must include reference to **real roots*.

\(-16<0\) so no real roots.

The quadratic formula is provided on the formula sheet.

Begin by stating the values of \(a\), \(b\) and \(c\).

\(x=0.9,x=-2.4\)

Take care to substitute negative values in brackets.

If you are asked to solve a quadratic equation in an exam and get a math error in your calculator, check your working carefully, expecially for any negative values.

\(x=1.4,x=0.4\)


Vectors

Substitute and evaluate. Remember that vectors are not fractions, so there should be no concerns with the "bottom numbers" being different.

\(\begin{pmatrix}19\\0\end{pmatrix}\)

Take care with negatives.

\(\begin{pmatrix}-2\\\phantom{-}2\\-6\end{pmatrix}\)

Finding the magnitude of \(\mathbf{u}=\begin{pmatrix}x\\y\end{pmatrix}\) requires the formula:

\(|\mathbf{u}|=\sqrt{x^2+y^2}\)

\(10\)

Finding the magnitude of \(\mathbf{u}=\begin{pmatrix}x\\y\\z\end{pmatrix}\) requires the formula:

\(|\mathbf{u}|=\sqrt{x^2+y^2+z^2}\)

\(\sqrt{35}\)

For (a), start by expressing the journey from P to R as a point-to-point journey:

\(\overrightarrow{PR}=\overrightarrow{PQ}+\overrightarrow{QR}\)

Then substitude the vectors required.

For (b), use the answer from part (a).

What is vector \(\overrightarrow{RS}\) equal to?

  1. \(\mathbf{u-v}\)

  2. \(\frac{1}{2}\mathbf{u-v}\)

Use the steps outlined in the previous question.

  1. \(\mathbf{a-b}\)

  2. \(\frac{1}{3}\mathbf{a}+\frac{2}{3}\mathbf{b}\)

For (a), think about the \(x\), \(y\) and \(z\) coordinates one-by-one.

Note that M is above the centre of the pyramid.

For (b), start by finding a vector to describe the journey from \(M\) to \(G\).

The length of a vector is given by its magnitude.

  1. M\((1.5,1.5,5)\)

  2. \(2.92\) units

Start by writing down each vector in component form.

\(\begin{pmatrix}5\\1\end{pmatrix}\)